How can a couple deal with the husband's non-Muslim mother who rejects and is jealous of the wife, while maintaining the husband's dutifulness towards his mother?

The mother is commanded to be honored and treated with kindness, even if she is a non-Muslim. This is based on the Almighty’s saying: "But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them, but accompany them in [this] world with appropriate kindness," and the Prophet’s (peace and blessings be upon him) saying to Asma bint Abi Bakr (may Allah be pleased with them both): "Maintain ties with your mother."

It is not permissible to obey the mother in a matter that is evil; obedience is only in what is good. One must deal with her tactfully and reach an understanding with the wife to avoid problems, especially since a mother’s jealousy of her son’s wife is a possibility. And his duties towards his wife are similar to her duties towards him, as the Almighty says: "And due to them [wives] is similar to what is expected of them, according to what is reasonable."